Existence and characterization of ground states for fourth order nonlinear Schrödinger systems

Pablo Álvarez-Caudevilla; Tatsuya Watanabe

doi:10.3934/dcds.2025070

Article Contents

2025, Volume 45, Issue 12: 4630-4645. Doi: 10.3934/dcds.2025070

This issue Previous Article Nonexistence results for quasilinear elliptic problems with singular lower order terms Next Article Aubry set of eikonal Hamilton–Jacobi equations on networks

Existence and characterization of ground states for fourth order nonlinear Schrödinger systems

Pablo Álvarez-Caudevilla^1, , and
Tatsuya Watanabe^2,

1.
Universidad Carlos Ⅲ de Madrid, Av. Universidad 30, 28911-Leganés, Spain
2.
Department of Mathematics, Faculty of Science, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-ku, Kyoto-City, 603-8555, Japan

^*Corresponding author: Pablo Álvarez-Caudevilla
^*Corresponding author: Pablo Álvarez-Caudevilla

Received: January 2024

Revised: May 2025

Early access: May 23, 2025

Published online: May 23, 2025

The second author is supported by JSPS KAKENHI Grant Numbers JP21K03317, JP24K06804.

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Abstract

The purpose of this paper is to establish the existence of ground states without restricting ourselves to the space of radial functions for a class of fourth order nonlinear Schrödinger systems of the form:

$ \left\{\begin{array}{ll} \Delta^2 u_1 - \alpha_1 \Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{2}u_1 + \beta |u_2|^{2}u_1 \\ \Delta^2 u_2 - \alpha_2 \Delta u_2 + \lambda_2 u_2 = \mu_2 |u_2|^{2}u_2 + \beta |u_1|^{2}u_2 \end{array} \right. \quad \hbox{in} \ \mathbb{R}^N. $

For our particular problem, neither the maximum principle nor the Schwarz symmetric rearrangement can be applied, as is usually performed for second order problems. Consequently, in order to arrive at the results obtained here, we remove the radial symmetry by applying the Fourier rearrangement, following a totally different approach from the usual one used for these types of problems. A classification of whether the ground states are semi-trivial or fully-nontrivial is also presented, obtaining a precise analysis depending on the parameters involved in the system. Our results complement those given by Álvarez-Caudevilla, Colorado and Galaktionov in [1] for the case $ \alpha_j = 0 $ $ (j = 1, 2) $.

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Mathematics Subject Classification: Primary: 35J91, 35J50; Secondary: 35Q55, 35B38.

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Figure 1. Classification of ground states of (1.1) with $ \Lambda = \min\{\sigma_1, \sigma_2\} $ and $ \Lambda' = \max\{\sigma_1, \sigma_2\} $

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References

[1]	P. Álvarez-Caudevilla, E. Colorado and V. A. Galaktionov, Existence of solutions for a system of coupled nonlinear stationary bi-harmonic Schrödinger equations, Nonlinear Analysis RWA, 23 (2015), 78-93. doi: 10.1016/j.nonrwa.2014.11.009.
[2]	A. Ambrosetti and E. Colorado, Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Acad. Sci. Paris Ser. I, 342 (2006), 453-458. doi: 10.1016/j.crma.2006.01.024.
[3]	N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams, Champman & Hall, London, 1997.
[4]	G. Baruch, G. Fibich and E. Mandelbaum, Singular solutions of the biharmonic Nonlinear Schrödinger equation, SIAM J. Appl. Math., 70 (2010), 3319-3341. doi: 10.1137/100784199.
[5]	D. Bonheure, J. B. Casteras, E. M. dos Santos and R. Nascimento, Orbitally stable standing waves of a mixed dispersion nonlinear Schrödinger equation, SIAM J. Math. Anal., 50 (2018), 5027-5071. doi: 10.1137/17M1154138.
[6]	D. Bonheure and R. Nascimento, Waveguide solutions for a nonlinear Schrödinger equation with mixed dispersion, Contributions to Nonlinear Elliptic Equations and Systems, Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, Cham, 86 (2015), 31-53. doi: 10.1007/978-3-319-19902-3_4.
[7]	H. Brascamp, E. Lieb and J. Luttinger, A general rearrangement inequality for multiple integrals, J. Funct. Anal., 17 (1974), 227-237. doi: 10.1016/0022-1236(74)90013-5.
[8]	A. Burchard, Cases of equality in the Riesz rearrangement inequality, Annals of Math., 143 (1996), 499-527. doi: 10.2307/2118534.
[9]	R. Carles and E. Moulay, Higher order Schrödinger equations, J. Phys. A: Math. Theor., 45 (2012), 395304, 11 pp. doi: 10.1088/1751-8113/45/39/395304.
[10]	J. Chabrowski and J. M. do Ó, On some fourth-order semilinear elliptic problems in $ \mathbb{R}^N$, Nonl. Anal., 49 (2002), 861-884. doi: 10.1016/S0362-546X(01)00144-4.
[11]	S. Correia, F. Oliveira and H. Tavares, Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with $d \ge 3$ equations, J. Funct. Anal., 271 (2016), 2247-2273. doi: 10.1016/j.jfa.2016.06.017.
[12]	B. Feng, J. Liu, H. Niu and B. Zhang, Strong instability of standing waves for a fourth-order nonlinear Schrödinger equation with the mixed dispersion, Nonlinear Anal., 196 (2020), 111791, 16 pp. doi: 10.1016/j.na.2020.111791.
[13]	A. Fernández, L. Jeanjean, R. Mandel and M. Mariş, Non-homogeneous Gagliardo-Nirenberg inequalities in ${{\bf{R}}}^N$ and application to a biharmonic non-linear Schrödinger equation, J. Differential Equations, 330 (2022), 1-65. doi: 10.1016/j.jde.2022.04.037.
[14]	G. Fibich, B. Ilan and G. Papanicolau, Self-focusing with fourth-order dispersion, SIAM J. Appl. Math., 62 (2002), 1437-1462. doi: 10.1137/S0036139901387241.
[15]	M. Grillakis, J. Shatah and W. Strauss, Stability theory of solitary waves in the presence of symmetry. Ⅰ, J. Funct. Anal., 74 (1987), 160-197. doi: 10.1016/0022-1236(87)90044-9.
[16]	E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, Int. Math. Res. Not. IMRN, 19 (2021), 15040-15081. doi: 10.1093/imrn/rnz274.
[17]	E. Lenzmann and T. Weth, Symmetry breaking for ground states of biharmonic NLS via Fourier extension estimates, J. Anal. Math., 152 (2024), 777-800. doi: 10.1007/s11854-023-0311-2.
[18]	T. C. Lin and J. Wei, Ground state of $N$ coupled nonlinear Schrödinger equations in $ \mathbb{R}^n$, $n\leq 3$, Comm. Math. Phys., 255 (2005), 629-653. doi: 10.1007/s00220-005-1313-x.
[19]	C. Liu and Z. Q. Wang, A complete classification of ground-states for a coupled nonlinear Schrödinger system, Commun. Pure Appl. Anal., 16 (2017), 115-130. doi: 10.3934/cpaa.2017005.
[20]	R. Mandel, Minimal energy solutions for cooperative nonlinear Schrödinger systems, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 239-262. doi: 10.1007/s00030-014-0281-2.
[21]	L. Maia, E. Montefusco and B. Pellacci, Positive solutions for a weakly coupled nonlinear Schrödinger system, J. Differential Equations, 229 (2006), 743-767. doi: 10.1016/j.jde.2006.07.002.
[22]	F. Natali and A. Pastor, The fourth-order dispersive nonlinear Schrödinger equation: Orbital stability of a standing wave, SIAM J. Appl. Dyn. Syst., 14 (2015), 1326-1347. doi: 10.1137/151004884.
[23]	M. K. Kwong, Uniqueness of positive solutions of $\Delta u- u+ u^p=0$ in $\mathbb{R}^N$, Arch. Rat. Mech. Anal., 105 (1989), 243-266. doi: 10.1007/BF00251502.
[24]	B. Pausader, The cubic fourth order Schrödinger equation, J. Funct. Anal., 256 (2009), 2473-2517. doi: 10.1016/j.jfa.2008.11.009.
[25]	T. Sato and T. Watanabe, Singular positive solutions for a fourth order elliptic problem in $ \mathbb{R}^N$, Commun. Pure Appl. Anal., 10 (2011), 245-268. doi: 10.3934/cpaa.2011.10.245.
[26]	A. Szulkin and T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, (2010), 597-632.
[27]	B. Sirakov, Least energy solitary waves for a system of nonlinear Schrödinger equations in $\mathbb{R}^n$, Comm. Math. Phys., 271 (2007), 199-221. doi: 10.1007/s00220-006-0179-x.
[28]	J. Wei and Y. Wu, Ground states of nonlinear Schrödinger systems with mixed couplings, J. Math. Pures Appl., 141 (2020), 50-88. doi: 10.1016/j.matpur.2020.07.012.
[29]	H. F. Weinberger, Variational methods for eigenvalue approximation, Regional Conference Series in Applied Mathematics, 15 (1994).